In my 15+ years of piano lessons from childhood through college, I always heard/said it "minnit" because it's fast-paced and quick to play--it takes about a minute. Not exactly, and no one times you, but it's such a catchy nickname and captures the feeling of it nicely. In books of piano music it's usually the shortest and quickest piece in the book, so why not? The 100 Years War wasn't exactly 100 years, was it?
Later, I heard the "my newt" theory and I thought...well, fine, that makes sense too. But the only time I hear anyone actually say "the my newt waltz" is when someone says, snobbily, "actually, it's pronounced 'my newt.'" My musician friends always say "minnit."
What difference does it make, anyways? I think both ways fit, and it's fine when words have multiple meanings.